Quaternionic LVM Manifolds

Abstract

An admissible configuration $Λ\subset\Hh^m\simeq\Rr^{4m}$ defines a compact intersection of quadrics $Z_{\Hh}(Λ)$ and a free diagonal $\Sp(1)$ quotient $N_{\Hh}(Λ)$. We realize this quotient as the leaf space of a real dilation action on an open subset of quaternionic projective space. The total space is a quaternionic moment-angle polyhedral product. Every quotient is $2$-connected; when there are no indispensable coordinates, it is $3$-connected and its fourth integral cohomology is generated by the Euler class $e$ of the associated oriented rank-four bundle. We compute $p_1(TN_{\Hh})=2(n-2)e$ and prove that no LVMQ manifold admits a quaternionic Kähler metric. An explicit infinite family consists of products of spheres admitting both homogeneous positive Einstein metrics and quaternionic toric structures in the sense of Gentili--Gori--Sarfatti. We also describe the trace foliations of the real action in the Poincaré domain. For the separate rank-one distribution $X\Hh$, an invertible quaternionic-linear field is integrable only in the real scalar case. A nonlinear field with positive real scalar derivative at the origin and involutive $X\Hh$ has exactly the standard quaternionic Hopf foliation on every sufficiently small centered sphere.

Revised version. Several proofs have been corrected and expanded, the exposition reorganized, and the references updated. 41 pages